11.13 化合物利息

章节大纲

• 选择节常规

  折叠 展开

  常规

  全部折叠 展开全部

  • 选择活动新闻通告

    

    新闻通告 讨论区
• 选择节化合物利息

  折叠 展开

  化合物利息

  播放段落

  Many financial transactions , including savings accounts, loans, credit card interest, and investments, involve a type of interest called compound interest . For example, say you deposit $1,000 into a savings account that pays 2.5% annual interest, and you would like to know the balance after three years if the interest is compounded, or calculated, monthly.
  ::许多金融交易,包括储蓄账户、贷款、信用卡利息和投资,都涉及一种称为复利的利息。 比如说,您将1 000美元存入一个每年支付2.5%利息的储蓄账户,如果利息是每月加起来或计算的,您想知道三年后的余额。

  播放段落

  With our knowledge of , we can now consider compound interest below. 
  ::有了我们的知识,我们现在可以考虑下面的复合兴趣。

  

  播放段落

  What Is Compound Interest? 
  ::什么是复利?

  播放段落

  Many financial accounts involve a type of interest called compound interest. Until now, we have calculated , a percentage rate of an amount over a period of time.  
  ::许多财务账户涉及一种称为复利的利息。 到目前为止,我们计算了一个时期的百分率,即一个时期的金额。

  播放段落

     Simple Interest Formula
  ::简单利息公式

  播放段落

  $$\begin{array}{r}I=Prt\end{array}$$

  
  $\begin{array}{r}I\end{array}$  is the interest earned.
  ::I=Prt I是所得利息。

  播放段落

  $\begin{array}{r}P\end{array}$  is the principal or initial investment.
  ::P是主要投资或初始投资。

  播放段落

  $\begin{array}{r}r\end{array}$  is the annual interest rate.
  ::r 是年利率。

  播放段落

  $\begin{array}{r}t\end{array}$  is time in years.
  ::t 是年中的时间 。

  播放段落

  Example 1
  ::例1

  播放段落

  An account calculates the interest on your investment of $1,000 after one year. The annual interest rate for the account is 4%. How much interest did you earn? How much money do you have after one year?
  ::一个账户计算一年后您投资的1 000美元的利息。 该账户的年利率为4%。 您赚了多少利息? 一年后您有多少钱?

  播放段落

  Solution: First we need to identify the principal, the annual interest rate, and the time to calculate the simple interest.
  ::解决方案:首先,我们需要确定本金、年利率和计算简单利息的时间。

  播放段落

  $$\begin{array}{rl}P& =1,000\\ r& =4\mathrm{\%}=0.04\\ t& =1\\ \\ I& =Prt=1,000\cdot 0.04\cdot 1=40\end{array}$$

  
  ::P=1 000r=40.04t=1I=Prt=1 0000.041=40

  播放段落

  The account earned $40 in interest. Together, you have the $1,000 you invested and the $40 in interest, or $1,040.
  ::该账户赚取了40美元的利息。加在一起,你们投资了1 000美元,利息40美元,即1 040美元。

  播放段落

  Usually, interest is not calculated once a year. Depending on the application, it can be calculated quarterly, monthly, or even daily. When this happens, the annual interest rate needs to be divided evenly among the number of times of year the interest is calculated. Also, the interest would be calculated on a new amount. For example, in the 2nd year of the investment in Example 1, there is a new principal, $1,040. Let's consider how calculating the interest more than once a year  works in another example.    
  ::通常情况下, 利息不是每年计算一次。 取决于应用程序, 它可以按季度、 月甚至每天计算。 发生这种情况时, 年利率需要平均地分配于利息计算的年数。 另外, 利息将按新的金额计算 。 例如, 在例1投资的第二年, 例1 中, 有一个新的本金, 1 040 美元。 我们来考虑另一个例子, 如何每年一次以上计算利息 。

  播放段落

  Example 2
  ::例2

  播放段落

  The investment account in Example 1 chooses to calculate interest four times a year instead of once a year. If you invest $1,000 with an annual interest rate of 4%, how much money will you earn after one year?
  ::例1中的投资账户选择每年四次而不是每年一次计算利息。 如果您投资1 000美元,年利率为4%,那么一年后你将挣多少钱?

  播放段落

  Solution: Organizing this in a  table will be helpful. We also need to divide the annual interest rate among the four periods when the interest will be calculated, $\begin{array}{r}t=\frac{1}{4}\end{array}$ . The account will earn 1% interest each period. 
  ::解决方案 : 将这个设置在表格中会有帮助 。 我们还需要将年利率分为四个计算利息的时期, t=14。 账户将每个时期赚取1%的利息 。

  Times Interest Calculated	Interest	Total Amount in Account After Interest Calculated
  1	$\begin{array}{r}I=1,000\cdot 0.04\cdot \frac{1}{4}=10\end{array}$	$1,010
  2	$\begin{array}{r}I=1,010\cdot 0.04\cdot \frac{1}{4}=10.10\end{array}$	$1,020.10
  3	$\begin{array}{r}I=1,020.10\cdot 0.04\cdot \frac{1}{4}=10.201\end{array}$	$1,030.30
  4	$\begin{array}{r}I=1,030.30\cdot 0.04\cdot \frac{1}{4}=10.303\end{array}$	$1,040.60

  播放段落

  After one year, there is $1,040.60 in the account.
  ::一年之后,账户中有1 040.60美元。

  播放段落

  It would be helpful to be able to determine the amount at the end of the year without having to find all the values in between. Let's try to generalize this process.
  ::如果能够在年底确定数额,而不必在中间找到所有数值,那将会很有帮助。让我们试着推广这一进程。

  播放段落

  Investigation: Finding the Compound Interest Formula
  ::调查:寻找复合利益公式

  播放段落

  1. If t he interest is calculated  $\begin{array}{r}n\end{array}$  times per year, find the time for one period where the interest is calculated:  $\begin{array}{r}t=\end{array}$  ______________
  ::1. 如果利息是每年n次计算的,应找到计算利息的某一时期的时间: t= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

  播放段落

  2. Find the interest in one period:  $\begin{array}{r}I=Pr\end{array}$ ___________
  ::2. 确定一个时期的利息:I=Pr

  播放段落

  3. Find the total amount after one period: $\begin{array}{r}A=P+\end{array}$  __________
  ::3. 确定一个时期之后的总金额:A=P+___________________________

  播放段落

  4. Now, find the amount  in a 2nd period using the amount in step 3 by adding the new amount to the interest on the new amount:  _____________________
  ::4. 现在,用第3步中的金额来计算第二期的金额,在新金额的利息中加上新的金额:___________________________________________________________________________________________________

  播放段落

  5. Factor out the common factor in step 4. $\begin{array}{r}A=\end{array}$  _____________________
  ::5. 将共同因素纳入第4步A=__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

  播放段落

  6. Repeat this process for a 3rd period. Using the amount in step 5, find the amount in a 3rd period by adding the new amount to the interest on the new amount:______________________
  ::6. 在第三个期间重复这一程序。使用第5步中的金额,在第三个期间通过在新金额的利息中加上新金额来找到该金额:_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

  播放段落

  7. Factor out the common factor in step 6. $\begin{array}{r}A=\end{array}$  _____________________ 
  ::7. 将共同因素纳入第6步A=_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

  播放段落

  Notice we keep getting powers of the same factor. 
  ::注意我们不断得到 同一因素的力量。

  播放段落

  To generalize this process, note you will calculate interest  $\begin{array}{r}n\cdot t\end{array}$  times over a period of  $\begin{array}{r}t\end{array}$  years.
  ::为了概括这个过程, 注意您会在 T 年中计算利息的数次 。

  播放段落

     Compound Interest Formula
  ::化合物利息公式

  播放段落

  $$\begin{array}{r}A=P(1+\frac{r}{n}{)}^{nt}\end{array}$$

  
  $\begin{array}{r}A\end{array}$  is the amount after  $\begin{array}{r}t\end{array}$  years.
  ::A=P(1+rn)nt A 是 T 年之后的金额 。

  播放段落

  $\begin{array}{r}n\end{array}$  is the number of times you compound , that is, calculate interest per year. 
  ::n 是您复合的倍数,即每年计算利息。

  播放段落

  $\begin{array}{r}P\end{array}$   is the principal.
  ::P是校长

  播放段落

  $\begin{array}{r}r\end{array}$   is the annual interest rate.
  ::r 是年利率。

  播放段落

  Here is some important vocabulary to know for compound interest: 
  ::以下是一些重要词汇,

  Number of Times Interest Is Calculated in a Year	Vocabulary
  1	annually
  2	semiannually
  4	quarterly
  12	monthly
  365	daily

  播放段落

  Let's consider an example using the formula.
  ::让我们考虑一个使用公式的例子。

  播放段落

  Example 3
  ::例3

  播放段落

  You deposit $1,000 into a savings account that pays 2.5% annual interest. Find the balance after 3 years if the interest rate is compounded a. annually, b. monthly, c. daily.
  ::您将 1000 美元 存入 储蓄 账户 , 该账户 支付 2.5% 的 年息 。 如果 年利率 复数为 a., b. 月利率为 b. 月利率为 c. 日利率, 则在 3 年后找到余额 。

  播放段落

  Solution: For part a, we will use $\begin{array}{r}A=1,000(1.025{)}^3=1,076.89\end{array}$ , as we would expect from Example 1.
  ::解决办法:在a部分,我们将使用A=1 000(1.025)3=1 076.89,正如我们从例1中预期的那样。

  播放段落

  But to determine the amount if it is compounded in amounts other than yearly, we need to use the formula.  For part b, $\begin{array}{r}n=12\end{array}$ .
  ::但是,要确定是否以年度以外的金额加在一起的金额,我们需要使用公式。对于 b, n=12 部分, 需要使用公式。

  播放段落

  $$\begin{array}{rl}A& =1,000{\left(1+\frac{0.025}{12}\right)}^{12\cdot 3}\\ & =1,000(1.002{)}^{36}\\ & =1,077.80\end{array}$$

  
  ::A=1 000(1+0.02512)123=1 000(1.00236=1 077.80)

  播放段落

  In part c, $\begin{array}{r}n=365\end{array}$ .
  ::部分c, n=365。

  播放段落

  $$\begin{array}{rl}A& =1,000{\left(1+\frac{0.025}{365}\right)}^{365\cdot 3}\\ & =,1000(1.000068{)}^{1095}\\ & =1,077.88\end{array}$$

  
  ::A=1,000(1+0.0253653653653=1000(1.0000681095=1,077.88)

  播放段落

  by Mathispower4u works through an example with the compound interest formula. 
  ::Mathispower4u通过一个复合利息公式的例子开展工作。

   

  播放段落

  Example 4
  ::例4

  播放段落

  The Wetakeyourmoola Credit Card Company charges an annual percentage rate (APR) of 21.99%, compounded monthly. If you have a balance of $2,000 on the card, what would the balance be after 4 years (assuming you do not make any payments)? If you pay $200 a month to the card, how long would it take you to pay it off? You may need to make a table to help you with the 2nd question.
  ::我们拿下你的Moola信用卡公司,按21.99%的年百分比收费,每月复数。如果卡上有2 000美元的余额,四年后余额会是多少(假设你没有支付任何款项)?如果你每月向卡支付200美元,你需要花多久才能付清?你可能需要做一张表格来帮助你解决第二个问题。

  播放段落

  Solution: Y ou need to use the formula $\begin{array}{r}A=P{\left(1+\frac{r}{n}\right)}^{nt},\end{array}$  where $\begin{array}{r}n=12\end{array}$ because the interest is compounded monthly.
  ::解答: 您需要使用公式 A=P(1+rn)n, 因为利息是每月复合的, 所以 n=12 。

  播放段落

  $$\begin{array}{rl}A& =2,000{\left(1+\frac{0.2199}{12}\right)}^{12\cdot 4}\\ & =2,000(1,018,325{)}^{48}\\ & =4,781.65\end{array}$$

  
  ::A=2 000(1+0.219912)124=2 000(1 018 325)48=4 781.65

  播放段落

  To determine how long it would take you to pay off the balance, you need to find how much interest is compounded in one month, subtract $200, and repeat. A table might be helpful. For each month after the 1st, we will use the equation  $\begin{array}{r}B=R{\left(1+\frac{0.2199}{12}\right)}^{12\cdot (\frac{1}{12})}=R(1.018325)\end{array}$ , where $\begin{array}{r}B\end{array}$ is the current balance, and $\begin{array}{r}R\end{array}$ is the remaining balance from the previous month. For example, in month 2, the balance (including interest) would be $\begin{array}{r}B=1,800{\left(1+\frac{0.2199}{12}\right)}^{12\cdot \left(\frac{1}{12}\right)}=1,800\cdot 1.08325=1,832.99\end{array}$ .
  ::为了确定您需要多久才能付清余额, 您需要找到一个月的利息加起来多少, 减去200美元, 并重复。 表格可能有用 。 对于第一个月之后的每个月, 我们将使用公式B=R(1+0. 219912)12( 112)=R( 1.018325), 其中B是当前余额, R是前一个月的余额。 例如, 在第二个月, 余额( 包括利息)将是 B=1 800(1+0. 219912)12( 112)=1 8001.08325=1 832.99。

  Month	1	2	3	4	5	6
  Balance	2,000	1,832.99	1,662.91	1,489.72	1,313.35	930.09
  Payment	200	200.00	200.00	200.00	200.00	200.00
  Remainder	$1,800	1,632.99	1,462.91	1,289.72	913.35	730.09
  	7	8	9	10	11
  Balance	790.87	640.06	476.69	299.73	108.03
  Payment	200.00	200.00	200.00	200.00	108.03
  Remainder	590.87	440.06	276.69	99.73	0

  播放段落

  It would take you 11 months to pay off the balance, and you'd pay $108.03 in interest, making your total payment $2,108.03.
  ::你要花11个月才能付清余额 你就要付108.03美元的利息 让你的付款总额达到2 108.03美元

  播放段落

  by HCCMathHelp shows some examples of using the compound interest formula and continuously compounded interest.   
  ::HCCMathHelp提供一些使用复合利息公式和持续增加利息的例子。

   

  播放段落

  To estimate compounding daily, some financial institutions use an model. This is called compounding continuously . 
  ::为了估计每天的复利,一些金融机构使用一种模型,称为连续复利。

  播放段落

   Compounding Continuously
  ::连续 连续

  播放段落

  $$\begin{array}{r}A=P{e}^{rt}\end{array}$$

    A  is the amount after  t  years.
  ::A=Pert A是 T 年之后的金额 。

  播放段落

  P  is the principal.
  ::P是校长

  播放段落

  r  is the annual interest rate.
  ::r 是年利率。

  播放段落

  Example 5
  ::例5

  播放段落

  Gianna opens a savings account with $1,000, and it accrues interest daily at a rate of 5%. What is the balance in the account after 6 years? Estimate the balance after 6 years by compounding continuously. 
  ::Gianna开立了一个1 000美元的储蓄账户,每天以5%的利率计息。六年后的账户余额是多少?通过连续增加来估计六年后的余额。

  播放段落

  Solution:  To find the amount in the account after 6 years, we use the compound interest formula with  $\begin{array}{r}n=365\end{array}$ . 
  ::解决方案:为了在6年后在账户中找到这笔金额,我们使用n=365的复合利息公式。

  播放段落

  $$\begin{array}{rl}A& =1,000{\left(1+\frac{0.05}{365}\right)}^{365\cdot 6}\\ & =1,000(1.000137{)}^{2190}\\ & =1,349.83\end{array}$$

  
  ::A=1 000(1+0.053653653656=1 000(1.0001372190=1 349.83)

  播放段落

  To estimate this with an exponential growth model, we have
  ::为了用指数增长模型来估计这一点,我们

  播放段落

   

  $$\begin{array}{rl}A& =1,000e^{0.05\cdot 6}\\ & =1,000e^{0.3}\\ & =1,349.86\end{array}$$

  
  ::A=1,000e0.056=1,000e0.3=1,349.86

  播放段落

  Feature: Why Do People Invest in the Stock Market?
  ::特长:为什么人们投资股市?

  播放段落

  by Peter Gregory
  ::彼得·格雷戈里

  播放段落

  Why is investing important? Why are people so fascinated by the stock market?
  ::为什么投资很重要? 为什么人们如此着迷于股市?

  播放段落

  Why It Matters
  ::为何重要

  播放段落

  Everybody works to earn money and save a little for later in life. The appeal of investing lies in the concept of compound growth , which relies on exponential functions.
  ::投资的吸引力在于复合增长的概念,它依赖指数函数。

  播放段落

  Investors buy a part of a business, or a share, expecting the business to grow and the share to increase in value. If you bought the share for $100, you might expect its value to go up by 10%, which means that at the end of the year it would be worth $110. You have earned $10. In the 2nd year, how much did you earn? Another $10? No, because you earned 10% not on $100, but on $110. You earned on the earnings of the previous year. This is referred to as compounding .
  ::投资者购买一部分企业或股份,希望企业增长,份额增加。如果你以100美元买入股份,你可能会期望其价值增加10%,这意味着在年底价值将达到110美元。你赚了10美元。在第二年,你挣了多少钱?另外10美元?不,因为你挣了10%,不是100美元,而是110美元。你从上一年的收入中挣到了10%。这被称为“加价 ” 。

  播放段落

  If you kept the share for 30 years and it hadn’t compounded, it would be worth $400 by then. However, if it had compounded every year at 10%, it would be worth about $1,750! People invest in the stock market hoping to make lots of money.
  ::如果你将股权保留30年,而它没有变本加厉,那么到那时它的价值将是400美元。 但是,如果它每年增加10%,它的价值将是1,750美元。 人们在股市上投资,希望赚很多钱。

  

  播放段落

  Unfortunately, in real life, not every company grows, and not every company grows every year, making the stock market much more complicated.
  ::不幸的是,在现实生活中,并不是每家公司都在增长,也不是每家公司每年增长,使股票市场更加复杂。

  播放段落

    by Your Plan B Income demonstrates the power of compounding. 
  ::以B计划的收入证明 复合的力量。

   

  播放段落

  Summary
  ::摘要

  播放段落
  • To calculate compound interest, use the formula $\begin{array}{r}A=P(1+\frac{r}{n}{)}^{nt},\end{array}$  where $\begin{array}{r}A\end{array}$  is the amount after $\begin{array}{r}t\end{array}$  years,  $\begin{array}{r}r\end{array}$  is the annual interest rate, and $\begin{array}{r}n\end{array}$  is the number of times you compound per year.
    ::要计算复合利息,请使用公式A=P(1+rn)nt,其中A是年后数额,r是年利率,n是年复数。
  播放段落
  • To calculate interest compounded continuously, use an exponential growth model.   
    ::要连续计算利息,请使用指数增长模式。

  播放段落

  Review
  ::回顾

  播放段落

  1. If $12,000 is invested at 4% annual interest compounded monthly, how much will the investment be worth in 10 years? Give your answer to the nearest dollar.
  ::1. 如果12 000美元投资为每月4%的年息复利,十年内投资价值多少?回答最接近的美元。

  播放段落

  2. If $8,000 is invested at 5% annual interest compounded semiannually, how much will the investment be worth in 6 years? Give your answer to the nearest dollar.
  ::2. 如果每半年投资8 000美元,年利率为5%,再加上每半年投资一次,6年内投资价值多少?回答最接近的美元。

  播放段落

  3. If $20,000 is invested at 6% annual interested compounded quarterly, how much will the investment be worth in 12 years? Give your answer to the nearest dollar.
  ::3. 如果每年投资20 000美元,每年6%有兴趣的复合季度投资,12年的投资价值是多少?回答最近的美元。

  播放段落

  4. If $5,000 is invested at 8% annual interest compounded quarterly, how much will the investment be worth in 15 years? Give your answer to the nearest dollar.
  ::4. 如果以8%的年利率投资5 000美元,每季度复计,15年的投资价值是多少?回答最接近的美元。

  播放段落

  5. How much of an initial investment is required to insure an accumulated amount of at least $25,000 at the end of 8 years, at an annual interest rate of 3.75% compounded monthly? Give your answer to the nearest $100.
  ::5. 在8年结束时,按每月3.75%的年利率(复合利率为3.75%),保证累计金额至少25 000美元,最初投资需要多少资金?回答最接近的100美元。

  播放段落

  6. How much of an initial investment is required to insure an accumulated amount of at least $10,000 at the end of 5 years, at an annual interest rate of 5% compounded quarterly? Give your answer to the nearest $100.
  ::6. 在五年结束时,以5%的复合季度年利率累计至少10 000美元的累计金额保证需要多少初始投资?回答最接近的100美元。

  播放段落

  7. Your initial investment of $20,000 doubles after 10 years. If the bank compounds interest quarterly, what is your interest rate?
  ::7. 10年后,你最初投资20 000美元的双倍,如果银行每季增加利息,利率是多少?

  播放段落

  8. Ted invests $3,565 in a savings account earning 2.4% annual interest, compounded daily. What is the value of Terry's investment after 3 years? Round your answer to the nearest hundredth.
  ::8. Ted在一个储蓄账户投资3 565美元,每年赚取2.4%的利息,每天加起来,3年后Terry的投资价值是多少?

  播放段落

  9. Calvin invested $1,200 in a savings account that compounds semiannually. Eighteen months later, his balance is $1,310.15. Calvin does not remember his account's annual interest rate, but he saw a newspaper advertisement for Bank North that was offering a rate of 3.5% for new accounts. Should Calvin close out his existing account and open a new one at Bank North?
  ::9. Calvin在一个储蓄账户投资1 200美元,每半年增加一次。 18个月后,他的余额为1 310.15美元。 Calvin不记得他的账户年利率,但他看到北银行的报纸广告为新账户提供3.5%的利率。 卡尔文应该关闭他现有的账户,在北银行开设一个新账户吗?

  播放段落

  10. Laura is considering investing $5,600 in a continuously compounded account earning 3.6% annual interest for 2 ½ years. What will Laura's accumulation be after 2 ½ years?
  ::10. 劳拉正在考虑投资5 600美元在一个连续的复杂账户中,在2年半的时间里每年赚取3.6%的利息。 劳拉的积累在2年半之后会是什么?

  播放段落

  Answers for Review Problems
  ::回顾问题的答复

  播放段落

  Please see the Appendix.
  ::请参看附录。

  播放段落

  PLIX 
  ::PLIX

  播放段落

  Try this interactive that reinforces the concepts explored in this section: 
  
  ::尝试这一互动,强化本节所探讨的概念: